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landscape in heteropolymers is thus showing a dierent dimensionality when imaged
at dierent energy resolutions. Diusion on such a landscape will therefore be not
only slower but also more involved at very low temperatures, requiring a lengthy
wandering on an involved network of tiny wrinkles.
The scenario emerging from the analysis of the spectral properties of connec-
tivity graphs of these very simplied models (which nonetheless exhibit the same
qualitative kinds of behaviors of real proteins as the approach to the folded state is
considered) is that the zero{temperature connectivity graph, which contains no in-
formation at all about saddle height, is not able to discern any detail of amino{acidic
sequence of the system. As soon as some information about saddles is introduced in
the form of a nite temperature connectivity graph, the distinction between homo
and heteropolymers becomes evident and it is highlighted by a smaller spectral
dimension for the connectivity graph of heteropolymers. In this model the higher
frustration of homopolymers does indeed result in a more complicated energy land-
scape but only when looking at a coarse grained level the energy landscape: the
ne details are frustration independent.
6.2.3. Three-dimensional models
Let us now briey discuss some results recently obtained applying the techniques
described above to a fully three-dimensional o-lattice coarse-grained model for
short peptides that has been recently studied by Clementi and coworkers [7, 21]:
further details can be found in [22].
The analysis of the latter model is conrming many of the features described
in Sec. 6.2.2, while at the same time providing a glimpse on the phenomenological
variability of these systems. A totally new phenomenon that is detected is a gradual
transformation of the connectivity graph in a star as temperature approaches the
T
from below. Star graphs are graphs characterized by the presence of a unique
central node connected to many \leaves" which are themselves not linked to each
other. The central node is actually the fastest growing one and corresponds to the
minimal energy node.
Notably enough, in this model, the spectral dimension of the connectivity graph
is basically independent on temperature, at least before leaf nodes appear and the
graph starts its transformation in to a star. At variance with the two-dimensional
case, it does show already at T = 0 an interesting correlation with the properties
of the aminoacidic sequence. Actually for three analyzed sequences of the same
length, a homopolyer and two heteropolymers characterized by dierent folding
propensities, we notice that the spectral dimension of the bad folding heteropolymer
is comparable to that of the homopolymer ( d = 171 in the rst case and d =
161 in the latter). On the contrary the good folder is characterized by a much
smaller dimensionality: d = 10:30:6. In this three-dimensional model the spectral
dimension seems then to be an even more useful quantity than in the 2{d case to
discriminate between good and bad folders.
